Effects of Population Size, Mating System, and Evolutionary Origin on Genetic Diversity in Spiranthes sinensis and S. hongkongensis

Hong Kong once supported more than 109 species of wild orchids, of which approximately 30% were endemic. Most of the local wild orchids have now become rare or endangered. I conducted a comparative study of genetic diversity in two closely related terrestrial orchids, an allotetraploid, Spiranthes hongkongensis , and its diploid progenitor, S. sinensis , to assess the effects of the population bottleneck associated with the origin of the polyploid and to investigate the relationships between number of breeding individuals, mating system, and level of isozyme variation in their populations. Nearly complete genetic uniformity was observed both within and among populations of S. hongkongensis . In contrast, S. sinensis had high levels of genetic variation for all of the genetic parameters examined. Regression analysis of population size and several components of genetic diversity in S. sinensis revealed that, among various measures of within-population variation, the proportion of polymorphic loci ( P ) and average number of alleles per locus ( A ) or per polymorphic locus ( A _p ) were the most sensitive to population size ( R ² = 0.942, p = 0.001; R ² = 0.932, p = 0.002; and R ² = 0.923, p = 0.002 respectively). The highly negative correlation ( r = −0.999, p < 0.01) between population size and the mean frequency of private alleles in pairwise population comparisons, p (1), indicated that population size may also be used to predict the extent of population differentiation caused by random genetic drift. Conservation of genetic diversity in S. sinensis could be maximized by protecting several of both large and small populations, whereas fewer populations may be needed to achieve this goal for S. hongkongensis.     

Genetic Diversity in a Threatened Wetland Species, Helonias bullata (Liliaceae)

Genetic variation was examined in Helonias bullata , a threatened perennial plant species that occurs in isolated wetland habitats. Fifteen populations representing the species' geographic range were sampled. Genetic diversity was low for the species ( H _es = 0.053) as well as within populations ( H _ep = 0.029). Of the 33 allozyme loci examined, 11 (33%) were polymorphic, while on average only 12.8% (4) of the loci were polymorphic within populations. The number of alleles per polymorphic locus was 2.36 for the species and averaged 2.09 across populations. For every genetic parameter calculated, variation in H. bullata was lower than that typically found for narrowly distributed plant species. The lowest levels of genetic diversity were found in northern areas that were colonized following the last glacial epoch. The number of genotypes detected per population ranged from three to 21, with a mean of 13 for this clonally reproducing species. We found a relatively high proportion of total genetic diversity (30.6%) among populations and a significant correlation (p < 0.002) between genetic distance and geographic distance. Genetic drift phenomena appear to play a major role in the population genetics of this species. Anomalously, several populations that appeared most limited in size and vigor were genetically most variable, perhaps because they represent older, relictual populations. Life-history characteristics of H. bullata coupled with low levels of genetic diversity and the degradation and disappearance of wetlands threaten the existence of this species.     

Genetic Diversity in the Endangered Lily Harperocallis flava and a Close Relative, Tofieldia racemosa

We examined genetic diversity in 464 individuals of the monotypic lily Harperocallis flava in its two habitats (seepage bogs and a roadside right-of-way) and five populations of a co-occurring related lily, Tofieldia racemosa. The endangered H. flava, endemic to the Apalachicola lowlands of the Florida panhandle, was monomorphic for the 22 loci scored. In contrast, T. racemosa had a high proportion of polymorphic loci ( P_s = 68.2%; P_p = 47.7%) with moderate genetic diversity (   H_es = 0.134; H_ep = 0.114). Estimated gene flow was moderately high ( Nm = 2.07) for T. racemosa, with most (93%) of the total genetic diversity found within populations. Despite the low level of genetic divergence, some isolation by distance was detected among T. racemosa populations. Harperocallis flava and other species without discernable genetic variation pose special problems for conservation biologists because genetic criteria are not available for the development of ex situ and in situ conservation and management strategies.     

Comparisons of Genetic Diversity in the Endangered Adenophora lobophylla and Its Widespread Congener, A. potaninii

Abstract: Starch-gel electrophoresis was used to examine the levels and distribution of genetic diversity in two Adenophora species: the narrow endangered Adenophora lobophylla and its widespread congener, A. potaninii . Based on allozyme variation at 18 putative loci, we measured high levels of genetic variability both in the endangered and the widespread species, with 83.3% of the loci being polymorphic. The mean expected heterozygosity within populations (   H _ep  ) and within species (   H _es  ) were 0.234 and 0.244 for A. potaninii and were as high as 0.210 and 0.211 for A. lobophylla . There was higher differentiation among populations in A. potaninii (   F _ST = 0.155) than in A. lobophylla (   F _ST = 0.071). The high levels of genetic diversity in the present allozyme survey are consistent with the morphological variation observed in these species and may be attributed to high outcrossing rates in the Adenophora species. In addition, A. lobophylla was identified as a distinct species on the basis of Nei's genetic distances and thus should be given a high priority for protection. It is noteworthy that the endangered A. lobophylla maintains much higher genetic diversity than most endemic or narrowly distributed plant species in spite of its restricted distribution. We hypothesize that A. lobophylla has become endangered for ecological and stochastic reasons, including habitat destruction or environmental changes, mud slides, and human disturbance such as grazing and mowing. Consequently, habitat protection is of particular importance for conserving this endangered species.     

Population Demographics and Genetic Diversity in Remnant and Translocated Populations of Sea Otters

Abstract: The effects of small population size on genetic diversity and subsequent population recovery are theoretically predicted, but few empirical data are available to describe those relations. We use data from four remnant and three translocated sea otter ( Enhydra lutris ) populations to examine relations among magnitude and duration of minimum population size, population growth rates, and genetic variation. Mitochondrial (mt)DNA haplotype diversity was correlated with the number of years at minimum population size ( r _s = −0.741, p = 0.038) and minimum population size ( r _s = 0.709, p = 0.054). We found no relation between population growth and haplotype diversity, although growth was significantly greater in translocated than in remnant populations. Haplotype diversity in populations established from two sources was higher than in a population established from a single source and was higher than in the respective source populations. Haplotype frequencies in translocated populations of founding sizes of 4 and 28 differed from expected, indicating genetic drift and differential reproduction between source populations, whereas haplotype frequencies in a translocated population with a founding size of 150 did not. Relations between population demographics and genetic characteristics suggest that genetic sampling of source and translocated populations can provide valuable inferences about translocations.     

Unprecedented Low Levels of Genetic Variation and Inbreeding Depression in an Island Population of the Black-Footed Rock-Wallaby

Abstract: It has been argued that demographic and environmental factors will cause small, isolated populations to become extinct before genetic factors have a significant negative impact. Islands provide an ideal opportunity to test this hypothesis because they often support small, isolated populations that are highly vulnerable to extinction. To assess the potential negative impact of isolation and small population size, we compared levels of genetic variation and fitness in island and mainland populations of the black-footed rock-wallaby ( Petrogale lateralis [Marsupialia: Macropodidae]). Our results indicate that the Barrow Island population of P. lateralis has unprecedented low levels of genetic variation (  H _e = 0.053, from 10 microsatellite loci) and suffers from inbreeding depression (reduced female fecundity, skewed sex ratio, increased levels of fluctuating asymmetry). Despite a long period of isolation ( ∼ 1600 generations) and small effective population size (  N _e ∼ 15), demographic and environmental factors have not yet driven this population to extinction. Nevertheless, it has been affected significantly by genetic factors. It has lost most of its genetic variation and become highly inbred (  F _e = 0.91), and it exhibits reduced fitness. Because several other island populations of P. lateralis also exhibit exceptionally low levels of genetic variation, this phenomenon may be widespread. Inbreeding in these populations is at a level associated with high rates of extinction in populations of domestic and laboratory species. Genetic factors cannot then be excluded as contributing to the extinction proneness of small, isolated populations.     

Effects of Habitat Fragmentation on Effective Population Size in the Endangered Rio Grande Silvery Minnow

Abstract:  We assessed spatial and temporal patterns of genetic diversity to evaluate effects of river fragmentation on remnant populations of the federally endangered Rio Grande silvery minnow ( Hybognathus amarus ). Analysis of microsatellite and mitochondrial DNA detected little spatial genetic structure over the current geographic range, consistent with high gene flow despite fragmentation by dams. Maximum-likelihood analysis of temporal genetic data indicated, however, that present-day effective population size ( N_eV ) of the largest extant population of this species was 78 and the ratio of effective size to adult numbers ( N_eV/N ) was ∼ 0.001 during the study period (1999 to 2001). Coalescent-based analytical methods provided an estimate of historical (river fragmentation was completed in 1975) effective size ( N_eI  ) that ranged between 10⁵ and 10⁶. We propose that disparity between contemporary and historical estimates of N_e and low contemporary N_e/N result from recent changes in demography related to river fragmentation. Rio Grande silvery minnows produce pelagic eggs and larvae subject to downstream transport through diversion dams. This life-history feature results in heavy losses of yearly reproductive effort to emigration and mortality, and extremely large variance in reproductive success among individuals and spawning localities. Interaction of pelagic early life history and river fragmentation has altered demographic and genetic dynamics of remnant populations and reduced N_e to critically low values over ecological time.     

Predicting the Deleterious Effects of Mutation Load in Fragmented Populations

Abstract:  Human-induced habitat fragmentation constitutes a major threat to biodiversity. Both genetic and demographic factors combine to drive small and isolated populations into extinction vortices. Nevertheless, the deleterious effects of inbreeding and drift load may depend on population structure, migration patterns, and mating systems and are difficult to predict in the absence of crossing experiments. We performed stochastic individual-based simulations aimed at predicting the effects of deleterious mutations on population fitness (offspring viability and median time to extinction) under a variety of settings (landscape configurations, migration models, and mating systems) on the basis of easy-to-collect demographic and genetic information. Pooling all simulations, a large part (70%) of variance in offspring viability was explained by a combination of genetic structure ( F_ST ) and within-deme heterozygosity ( H_S ). A similar part of variance in median time to extinction was explained by a combination of local population size ( N ) and heterozygosity ( H_S ). In both cases the predictive power increased above 80% when information on mating systems was available. These results provide robust predictive models to evaluate the viability prospects of fragmented populations.     

Genetic Effects of Habitat Fragmentation on Common Species of Swiss Fen Meadows

Abstract:  The area of Caricion davallianae alliance in Switzerland has been considerably reduced and fragmented during the last 150 years. We assessed the genetic variability, inbreeding level, and among-population differentiation of two common habitat-specific plant species, Carex davalliana SM. and Succisa pratensis Moench, in 18 Caricion davallianae fen meadows subjected to fragmentation. We used a spatial field design of fen systems (six systems total), each consisting of one large habitat island and two small habitat islands. We used allozyme electrophoresis to derive standard genetic parameters ( A, P, H_O, H_E, F_IS, F_ST ). In Carex we identified a consistently lower A in isolated habitat islands; furthermore, H_E was lower in small habitat islands than in large habitat islands. In Succisa we identified a lower H_O in small habitat islands than in larger ones. Small habitat islands were marginally significantly differentiated (  F_ST ) from large islands for Succisa . For both species, no effects were evident for F_IS ; therefore, we argue that genetic drift rather than inbreeding is the main cause of the observed differences. The genetic structure of Carex and Succisa in small habitat islands differed from that in large habitat islands, but differences were small. It appears that the observed differences in genetic variability among fen meadows correspond to observed differences in fitness and demographic traits. We show that habitat fragmentation affects not only the rare species in an ecosystem but also reduces the survival probabilities of common species. One of the main goals of conservation should be to mitigate fragmentation of natural habitats in order to increase population sizes and connectivity.     

Lineage Loss in Serengeti Cheetahs: Consequences of High Reproductive Variance and Heritability of Fitness on Effective Population Size

Abstract: In natural populations, many breeders do not leave surviving offspring, and as a result many potential genetic lineages are lost. I examined lineage extinction in Serengeti cheetahs ( Acinonyx jubatus ) and found that 76% of matrilines were lost over a 25-year period. Production of future breeders was nonrandom and generally confined to a few families. Five out of 63 matrilines accounted for 45% of the total cheetah population over the course of the study. Lineage persistence is perhaps best illustrated by the variance in lifetime reproductive success ( LRS) and heritability in this parameter. In female cheetahs, variance in LRS was high, and new data show that this LRS was heritable. Variance in LRS and heritability in LRS have dramatic consequences for effective population size, N _e. I calculated N _e for cheetahs, taking into account fluctuating population size, unequal sex ratio, non-Poisson distribution of reproductive success, and heritability of fitness. The N _e was most strongly affected by variance in reproductive success and especially heritability in reproductive success. The variance N _e was 44% of the actual population size, and the inclusion of heritability further reduced N _e to only 15% of the actual population, a ratio similar to that of a social carnivore with reproductive suppression. The current cheetah population in the Serengeti is below numbers suggested by N _e estimates as sufficient to maintain sufficient genetic diversity.     

Developing a Sampling Strategy for Baptisia arachnifera Based on Allozyme Diversity

We analyzed the amount and distribution of genetic variation in Baptisia arachnifera Duncan to develop a sampling strategy for ex situ research. Baptisia arachnifera is an endangered plant species endemic to the coastal plain of Georgia (U.S.) where all populations are within 16 km of each other. A reduction in numbers of individuals has been observed during the last 50 years. Baptisia arachnifera was polymorphic at 24% of the 37 loci examined with an average of 1.32 alleles per locus. The genetic diversity index was relatively low ( H_e = 0.097) as expected for endemic species. Populations were in Hardy-Weinberg equilibrium, suggesting that the species is outcrossing. Consistent with this conclusion is the observation that the majority (approximately 90%) of the genetic variation present in the species is found within individual populations. Indirect evidence of gene flow between populations was detected (   Nm = 2.35). The close proximity of the populations and the recent reduction in population sizes suggest that the populations surveyed may be fragments of a once more continuous gene pool. Based on the observed distribution of genetic diversity among populations (G_ST = 0.096), sampling two populations would capture 99% of the allozyme diversity surveyed. Allozyme data were used to determine which 2 of the 10 populations surveyed should be sampled to maximize the ex situ conservation of genetic diversity. Although the paper-producing companies that own most of the land where Baptisia arachnifera occurs are modifying their harvesting techniques, the species could become extinct without more effective management and preservation efforts.     

Effective Population Size and Maintenance of Genetic Diversity in Captive-Bred Populations of a Lake Victoria Cichlid

Abstract: We used microsatellite DNA markers to investigate the maintenance of genetic diversity within and between samples of subpopulations (spanning five captive-bred generations) of the haplochromine cichlid Prognathochromis perrieri . The subpopulations are maintained as part of the Lake Victoria Cichlid species survival plan. Changes in the frequencies of 24 alleles, over four polymorphic loci, were used to estimate effective population size (   N _e   ). Point estimates of N _e ranged from 2.5 to 7.7 individuals and were significantly smaller than the actual census size (   N _obs  ) for all subpopulations (32–243 individuals per generation), with the corresponding conservative N _e   /  N _obs ratios ranging from 0.01 to 0.12. Approximately 19% of the initial alleles were lost within the first four generations of captive breeding. Between-generation comparisons of expected heterozygosity showed significant losses ranging from 6% to 12% per generation. Seven private alleles were observed in the last sampled generation of four subpopulations, and analysis of population structure by F _ST indicated that approximately 33% of the total genetic diversity is maintained between the subpopulations from different institutions. To reduce the loss of genetic variation, we recommend that offspring production be equalized by periodically removing dominant males, which will encourage reproduction by additional males. Consideration should also be given to encouraging more institutions to maintain populations, because a significant fraction of the genetic variation exists as among-population differences resulting from random differentiation among subpopulations.     

Evaluating Genetic Diversity Associated with Propagation-Assisted Restoration of American Shad

Abstract: We investigated the conservation of genetic diversity during a restoration program for American shad ( Alosa sapidissima ) in Virginia ( U.S.A.). Restoration entailed capture of wild Pamunkey River shad broodstock followed by production and release of hatchery-reared fry to supplement the nearly extinct James River shad population. To assess the baseline genetic diversity of donor and recipient populations, we used five tri- and tetra-nucleotide microsatellite loci to test for genetic heterogeneity among yearly subsamples from both rivers and between early- and late-spawning shad from the donor population. Tests for allelic heterogeneity between James River and Pamunkey shad subsamples yielded no significant genetic differentiation (χ ² = 14.72, p = 0.132 and χ ² = 10.24, p = 0.440, respectively). We detected no significant genetic divergence between early- and late-spawning adults in Pamunkey River spawning aggregations in either year. The donor and recipient populations exhibited significant genetic differentiation (χ ² = 27.4, p = 0.003), however, indicating that the stocking program carries a risk of outbreeding depression. Because the two river populations are genetically divergent, replenishment of the James population with Pamunkey fry may be detectable in the future as heterozygote deficits and linkage disequilibria in the James River population. In an analysis of broodstock and their hatchery-reared progeny, microsatellites proved efficient for family analysis, unambiguously determining the parentage of 100% of the hatchery-reared fry studied. Genetic analysis indicated that breeding procedures may result in high levels of reproductive variance.     

Genetic Structure of a Mimosoid Tree Deprived of Its Seed Disperser, the Spider Monkey

Abstract: To assess the genetic consequences for a Neotropical tree of the loss of its main seed disperser, we compared the genetic structure of Inga ingoides in a site where the spider monkey (Ateles paniscus) was abundant and a site where it had been eliminated by subsistence hunting. Gene flow should be reduced in the site where the spider monkey is absent, and there should be a corresponding subpopulation differentiation of seedlings within the spatial range of the movements of these primates in the absence of between-site differences in allelic frequencies. At the microhabitat (  family) scale, seedlings growing under parent plants should be genetically more related in the absence of the spider monkey than in its presence. Subpopulation differentiation was smaller where the spider monkey was present (  four loci, F_ST = 0.011) than where it was absent (  four loci, F_ST = 0.053) for the first year of study, but not for the second year (three loci, F_ST = 0.005 vs. 0.003). The number of alleles in common among seedlings growing under parent plants was smaller in the presence of the spider monkey than in its absence, showing family genetic structure in the first generation for both years of study ( Mann-Whitney, z = −2.17, p = 0.03 and z = −2.72, p = 0.006 for 1996 and 1997, respectively). This family genetic structure in the first generation should accelerate the development of population genetic structure. Development of genetic structure might result in demographic changes, one of which would be a fitness reduction if the species were self-incompatible, as suggested for Inga by available evidence. Large birds and mammals are the main targets of subsistence hunting in the Neotropics. Extinction of seed-dispersing frugivores may result in pronounced changes in the demographic and genetic structure of tree species in Neotropical forests.     

Rapid Loss of Genetic Variation in Large Captive Populations of Drosophila Flies: Implications for the Genetic Management of Captive Populations

Levels of variation in eight large captive populations of D. melanogaster (census sizes ∼ 5000) that had been in captivity for periods from 6 months to 23 years (8 to 365 generations) were estimated from allozyme heterozygosities, lethal frequencies, and inversion heterozygosities and phenotypic variances, additive genetic variances ( V _A), and heritabilities ( h ²) for sternopleural bristle numbers. Correlations between all measures of variation except lethal frequencies were high and significant. All measures of genetic variation declined with time in captivity, with those for average heterozygosities, V _A, and h ² being significant. The effective population size ( N _e) was estimated to be 185–253 in these populations, only 0.037–0.051 of census size (N). Levels of allozyme heterozygosities declined rapidly in two large captive populations founded from another wild stock, being reduced by 86% and 62% within 2.5 years in spite of being maintained at sizes of approximately 1000 and 3500. Estimates of N _e/ N for these populations were only 0.016 and 0.004. Two estimates of N _e/ N for captive populations of D. pseudoobscura from data in the literature were also low at 0.036 and 0.012. Consequently, the rate of loss of genetic variation in captive populations and endangered species may be more rapid than hitherto recognized. Merely maintaining captive populations at large census sizes may not be sufficient to maintain essential genetic variation.     

Application of the One-Migrant-per-Generation Rule to Conservation and Management

Abstract:  Endangered species are commonly found in several (partially) isolated populations dispersed on different fragments of a habitat, natural reserve, or zoo. A certain level of connectivity among such populations is essential for maintaining genetic variation within and between populations to allow local and global adaptation and for preventing inbreeding depression. A rule of thumb widely accepted by the conservation community is that one migrant per generation (OMPG) into a population is the appropriate level of gene flow. This rule is based on Wright's study of his island model under a long list of simplifying assumptions. I examined the robustness of the OMPG rule to the violation of each of the many assumptions, quantifying the effect with population genetics theory. I showed that, when interpreted as one effective migrant per generation, OMPG is generally valid for real populations departing from the ideal model in the discrepancies of actual (  N ) and effective (  N_e  ) population sizes and actual ( m ) and effective ( m_e  ) migration rates. I also addressed the issue of converting the effective number of migrants (  M_e= N_em_e  ) into the actual number of migrants ( M = Nm  ) of a certain age and sex. In particular, N_e < N , a case common for natural populations, did not necessarily require M > M_e to maintain a certain level of differentiation among populations. Rather, translating the elusive M_e into the manageable M depends on the specific causes (e.g., biased sex ratio, reproductive skew) that lead to N_e < N .     

Relationship of Effective to Census Size in Fluctuating Populations

Abstract: The effective size of a population (    N _e   ) rather than the census size (    N ) determines its rate of genetic drift. Knowing the ratio of effective to census size, N _e  /   N , is useful for estimating the effective size of a population from census data and for examining how different ecological factors influence effective size. Two different multigenerational ratios have been used in the literature based on either the arithmetic mean or the harmonic mean in the denominator. We clarify the interpretation and meaning of these ratios. The arithmetic mean N _e  /   N ratio compares the total number of real individuals to the long-term effective size of the population. The harmonic mean N _e  /   N ratio summarizes variation in the N _e  /   N ratio for each generation. In addition, we show that the ratio of the harmonic mean population size to the arithmetic mean population size provides a useful measure of how much fluctuation in size reduced the effective size of a population. We discuss applications of these ratios and emphasize how to use the harmonic mean N _e  /   N ratio to estimate the effective size of a population over a period of time for which census counts have been collected.     

A Hierarchical View of Genetic Structure in the Rare Annual Plant Clarkia springvillensis

Genetic structure at several spatial scales was examined in the rare California annual, Clarkia springvillensis . Using seven isozyme-encoding loci as genetic markers, we assessed the amount and distribution of genetic variation among three populations and eight subpopulations. Total genetic variation was lower than in species with similar life history traits but equivalent to that of other endemic plants. Spatial autocorrelation showed some evidence for very limited differentiation within subpopulations at a scale of 1–2 m. The subpopulations, separated by tens of meters, were found to be more differentiated from each other ( F _sp = 0.084) on average than were populations ( F,pt = 0.017). This local genetic differentiation was not correlated with physical distance between subpopulations. The low F_pt estimates suggest that substantial gene flow is occurring among populations. However, the lack of correlation between genetic and geographic distances and the significant differentiation of subpopulations suggest that genetic drift is occurring within populations. Therefore, we believe the apparent homogeneity of populations is due to each population's gene frequencies' being an average of several divergent subpopulations. If drift is causing differentiation within populations, it may eventually cause differentiation between populations. The importance of using a hierarchical approach to evaluating genetic structure is clear. Patterns occurring at one spatial scale may not be evident at others. One should not necessarily conclude that gene flow is substantial and that the risk of genetic erosion via drift is negligible just because differentiation between populations is small; the system may not be at equilibrium. This lesson is particularly important when recent changes in climate or land use are apparent.     

Genetic Diversity of Lepilemur mustelinus ruficaudatus, a Nocturnal Lemur of Madagascar

The genetic polymorphism of natural populations of Lepilemur mustelinus ruficaudatus was studied by protein electrophoresis. We sampled blood from 72 individuals from four populations separated by geographic or anthropogenic barriers from southwestern Madagascar. Six out of 22 enzyme loci showed genetic variation with a degree of polymorphism of 0.273. The expected and observed degree of genetic heterozygosity over all loci is similar to that of other primates (H_e = 0.058, H_o = 0.036). The F-statistics revealed that the four subpopulations were similar with respect to gene structure (F_ST = 0.065, p = 0.016), but the genotypic structures within subpopulations were inconsistent with random mating. For the total of the four subpopulations the proportion of heterozygous individuals was significantly smaller than expected under random mating (F_IS = 0.373, F_IT = 0.414, p < 0.01). These results correspond closely to what is expected considering the low migration ability of individuals of L. m ruficaudatus leading to small and rather isolated inbred populations.     

Predicting Extinction Times from Environmental Stochasticity and Carrying Capacity

Managers of small populations often need to estimate the expected time to extinction T_e of their charges. Useful models for extinction times must be ecologically realistic and depend on measurable parameters. Many populations become extinct due to environmental stochasticity, even when the carrying capacity K is stable and the expected growth rate is positive. A model is proposed that gives T_e by diffusion analysis of the log population size n_t (= log_e N_t). The model population grows according to the equation N_t+1 = R_tN_t, with K as a ceiling. Application of the model requires estimation of the parameters k = logK, r_d = the expected change in n, v_r = Variance(log R), and ϱ the autocorrelation of the r_t. These are readily calculable from annual census data (r_d is trickiest to estimate). General formulas for T_e are derived. As a special case, when environmental fluctuations overwhelm expected growth (that is r_d 0), T_e = 2n_o(k - n_o/2)/v_r. If the r_t are autocorrelated, then the effective variance is v_re v_r (1 + ϱ)/(1 - ϱ). The theory is applied to populations of checkerspot butterfly, grizzly bear, wolf, and mountain lion.